6 A particle \(P\) starts at rest and moves in a straight line from a point \(O\). At time \(t\) s after leaving \(O\), the velocity of \(P , v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by \(v = b t + c t ^ { \frac { 3 } { 2 } }\), where \(b\) and \(c\) are constants. \(P\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 4\) and has velocity \(13.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 9\).

1. Show that \(b = 3\) and \(c = - 0.5\).
2. Find the acceleration of \(P\) when \(t = 1\).
3. Find the positive value of \(t\) when \(P\) is at instantaneous rest and find the distance of \(P\) from \(O\) at this instant.
4. Find the speed of \(P\) at the instant it returns to \(O\).